Periodic Spectral Problems

Periodic differential operators arise in many areas of physics and mathematics, and studying their spectral properties is very important. Spectra of periodic operators have a band-gap structure, that is, they consist of a collection of closed intervals possibly separated by gaps. There is a famous hypothesis, called the Bethe-Sommerfeld conjecture, which claims that the number of gaps is finite. It has been justified for the Schroedinger operator with an electric field. One aim of the present project is to prove the conjecture in a much more general setting. The solution is expected to require the use of the pseudo-differential calculus, geometry of lattices and geometrical combinatorics. An important quantitative characteristic of differential operators acting on a non-compact manifold in the so-called integrated density of states. This function is a natural analogue of the spectral counting function. We plan to study the behaviour of this function for large values of energy and, in particular, to prove that the density of states has a complete asymptotic expansion in the (negative as well as positive) powers of energy.We also plan to study more basic properties of the nature of the spectrum, namely whether the spectrum is absolutely continuous. We plan to give establish a wide range of sufficient conditions which guarantee the absolute continuity of the spectrum. Finally, we plan to study limit-periodic problems. These problems are natural generalisation of the periodic ones. While the class of limit-periodic operators is not as wide as the class of quasi-periodic or almost-periodic operators, some of the methods of the periodic theory are applicable to the limit-periodic case. We intend to prove the Bethe-Sommerfeld conjecture in the limit-periodic setting.